Intuition The big pictureAn orbit is a frozen ellipse floating in 3D space. To pin it down completely you need to answer six questions:
How big is the ellipse? → a a a How squashed is it? → e e e How tilted is its plane relative to a reference plane? → i i i Which way does the tilted plane swing around? → Ω \Omega Ω Which way does the ellipse point inside its own plane? → ω \omega ω Where is the satellite on the ellipse right now? → ν \nu ν M M M E E E
The first two (a , e a,e a , e shape . The next three (i , Ω , ω i,\Omega,\omega i , Ω , ω orientation . The last one (ν \nu ν position in time . That is exactly 6 numbers — and a position+velocity state ( r ⃗ , v ⃗ ) (\vec r,\vec v) ( r , v )
Definition State vector vs. orbital elementsA satellite's motion is fully fixed by its position r ⃗ \vec r r v ⃗ \vec v v 6 numbers . Orbital elements are an alternative set of 6 numbers that describe the same orbit, but in a way humans can picture geometrically and that mostly stay constant in the two-body problem.
WHAT is the point? In ( r ⃗ , v ⃗ ) (\vec r,\vec v) ( r , v ) five of the six (a , e , i , Ω , ω a,e,i,\Omega,\omega a , e , i , Ω , ω constant for an ideal Kepler orbit — only ν \nu ν
Definition Semi-major axisa a a size and, by Kepler's 3rd law, the period and total energy.
e e e squashed the ellipse is, 0 ≤ e < 1 0\le e<1 0 ≤ e < 1
e = 0 e=0 e = 0 e → 1 e\to 1 e → 1 e = 1 e=1 e = 1 e > 1 e>1 e > 1
a , e a,e a , e r p = 7000 r_p = 7000 r p = 7000 r a = 9000 r_a = 9000 r a = 9000
a = r p + r a 2 = 8000 a = \tfrac{r_p+r_a}{2} = 8000 a = 2 r p + r a = 8000 Why? The two extreme points span the full major axis 2 a 2a 2 a e = r a − r p r a + r p = 2000 16000 = 0.125 e = \tfrac{r_a - r_p}{r_a + r_p} = \tfrac{2000}{16000}=0.125 e = r a + r p r a − r p = 16000 2000 = 0.125 Why? The numerator is 2 a e 2ae 2 a e 2 a 2a 2 a e e e
i i i 0 ≤ i ≤ 180 ∘ 0\le i\le 180^\circ 0 ≤ i ≤ 18 0 ∘
i = 0 i=0 i = 0 prograde (orbits same way Earth spins).i = 90 ∘ i=90^\circ i = 9 0 ∘ polar (passes over both poles).i > 90 ∘ i>90^\circ i > 9 0 ∘ retrograde .
Ω \Omega Ω Ω \Omega Ω in the reference plane , from the vernal equinox Υ ^ \hat\Upsilon Υ ^ ascending node (the upward crossing point). It says which way the tilted plane is rotated about the z z z
Definition Argument of periapsisω \omega ω in the orbital plane , from the ascending node to periapsis (closest approach), in the direction of motion.
ν \nu ν only element that changes with time in a two-body orbit.
Recall Feynman: explain to a 12-year-oldImagine drawing an oval racetrack on a tilted dinner plate floating in space.
a a a e e e i i i Ω \Omega Ω ω \omega ω on the plate.ν \nu ν
Mnemonic Order = Shape → Orientation → Position"A Sweet Idea Of Wonderful Vistas" → a a a e e e i i i Ω \Omega Ω ω \omega ω ν \nu ν 2 shape, 3 orientation, 1 position.
Common mistake "Inclination tells you which way the plane is rotated."Why it feels right: both i i i Ω \Omega Ω The fix: i i i how steep the tilt is (angle from the equator). Ω \Omega Ω which compass direction that tilt faces (rotation about z z z i = 90 ∘ i=90^\circ i = 9 0 ∘ Ω \Omega Ω
a a a Why it feels right: for a circle, r = a r=a r = a The fix: a a a major axis of the ellipse, a constant; the actual distance r r r r p r_p r p r a r_a r a e = 0 e=0 e = 0 r = a r=a r = a
Common mistake "Energy depends on botha a a e e e Why it feels right: speed clearly changes with e e e The fix: ε = − μ / 2 a \varepsilon=-\mu/2a ε = − μ /2 a a a a only . Eccentricity reshuffles kinetic↔potential energy around the orbit but the total stays fixed.
ω \omega ω ν \nu ν Why it feels right: both are in-plane angles in the direction of motion.
The fix: ω \omega ω ν \nu ν u = ω + ν u=\omega+\nu u = ω + ν
Which element sets the orbital period and energy? The semi-major axis
a a a (since
ε = − μ / 2 a \varepsilon=-\mu/2a ε = − μ /2 a and
T = 2 π a 3 / μ T=2\pi\sqrt{a^3/\mu} T = 2 π a 3 / μ ).
Which orbital element changes with time in a two-body orbit? Only the true anomaly
ν \nu ν (position); the other five are constant.
Define eccentricity in words and give the 0 / 1 0/1 0/1 How squashed the ellipse is;
e = 0 e=0 e = 0 circle,
e → 1 e\to1 e → 1 parabola/escape.
Formula for a a a e e e r p , r a r_p,r_a r p , r a a = ( r p + r a ) / 2 a=(r_p+r_a)/2 a = ( r p + r a ) /2 ,
e = ( r a − r p ) / ( r a + r p ) \;e=(r_a-r_p)/(r_a+r_p) e = ( r a − r p ) / ( r a + r p ) .
What does inclination i i i Angle between the orbital plane and the equatorial reference plane.
What does i = 90 ∘ i=90^\circ i = 9 0 ∘ A polar orbit (passes over the poles).
What is the ascending node? The point where the satellite crosses the reference plane going south→north (upward).
What does RAAN Ω \Omega Ω Angle in the reference plane from vernal equinox to the ascending node (swing of the plane about
z z z ).
What does the argument of periapsis ω \omega ω In-plane angle from ascending node to periapsis along the motion.
Derive periapsis/apoapsis radii. r p = a ( 1 − e ) r_p=a(1-e) r p = a ( 1 − e ) ,
r a = a ( 1 + e ) r_a=a(1+e) r a = a ( 1 + e ) ; sum is
2 a 2a 2 a .
What vector points toward periapsis and has magnitude e e e The eccentricity vector
e ⃗ \vec e e .
How do you find inclination from a state vector? cos i = h z / ∣ h ⃗ ∣ \cos i=h_z/|\vec h| cos i = h z /∣ h ∣ with
h ⃗ = r ⃗ × v ⃗ \vec h=\vec r\times\vec v h = r × v .
Why does the velocity sign disambiguate ν \nu ν r ⃗ ⋅ v ⃗ > 0 \vec r\cdot\vec v>0 r ⋅ v > 0 → outbound (
0 < ν < 180 ∘ 0<\nu<180^\circ 0 < ν < 18 0 ∘ );
< 0 <0 < 0 → inbound.
Orbit (polar) equation with elements? r = a ( 1 − e 2 ) / ( 1 + e cos ν ) r=a(1-e^2)/(1+e\cos\nu) r = a ( 1 − e 2 ) / ( 1 + e cos ν ) .
The 3-2-1 split of the six elements? 2 shape (
a , e a,e a , e ), 3 orientation (
i , Ω , ω i,\Omega,\omega i , Ω , ω ), 1 position (
ν \nu ν ).
Vis-viva equation — gives v v v a , r a,r a , r ε = − μ / 2 a \varepsilon=-\mu/2a ε = − μ /2 a Kepler's Laws — third law ties a a a T T T Angular momentum in orbits — h ⃗ \vec h h i i i Ω \Omega Ω Eccentricity vector — defines e e e ω \omega ω State vector to orbital elements conversion — the full algorithm using these meanings.Orbital perturbations — why Ω , ω \Omega,\omega Ω , ω
State vector r and v
Orbital elements
Reference frame
Intuition Hinglish mein samjhoSocho ek orbit ek oval (ellipse) hai jo space me ek tilted plate par bani hai. Is poori cheez ko exactly describe karne ke liye humein 6 numbers chahiye — yahi orbital elements hain. Pehle do shape batate hain: a a a e e e a a a ε = − μ / 2 a \varepsilon = -\mu/2a ε = − μ /2 a
Agle teen numbers orientation batate hain — yaani plate kaise tilt aur rotate hui hai. i i i Ω \Omega Ω ω \omega ω i i i Ω \Omega Ω
Chhatha number ν \nu ν a , e , i , Ω , ω a,e,i,\Omega,\omega a , e , i , Ω , ω ν \nu ν ( r ⃗ , v ⃗ ) (\vec r,\vec v) ( r , v )
